Problem 51 - College Algebra

Question

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} \frac{3}{x^{2}}+\frac{1}{y^{2}}-7 \\ \frac{5}{x^{2}}-\frac{2}{y^{2}}--3 \end{array}\right.$$

Step-by-Step Solution

Answer

The system solution is \(x = 0\) and \(y = 0\).

Step 1 - Eliminate the fractions

Multiply each equation by their lowest common multiple (LCM) to get rid of the denominators. The LCM of \(x^{2}\) and \(y^{2}\) is \(x^{2}y^{2}\), multiplying each equation by \(x^{2}y^{2}\) will give:\[\begin{{align*}}3y^{2} + x^{2} - 7x^{2}y^{2} = 0, \5y^{2} + 2x^{2} - 3x^{2}y^{2} = 0.\end{{align*}}\]

Step 2 - Rearrange the first equation

Rearrange the first equation to isolate \(x^{2}\), which can be substituted in the second equation:\[x^{2} = \frac{3y^{2} - 7x^{2}y^{2}}{1} = 3y^{2}(1 - 7xy^{2}).\]

Step 3 - Substitute into the second Equation

Substitute \(x^{2}\) in second equation and simplify to get:\[\begin{{align*}}5y^{2} + 2x^{2} - 3x^{2}y^{2} = 0 &\Rightarrow 5y^{2} + 2[3y^{2}(1 - 7xy^{2})] - 3[3y^{2}(1 - 7xy^{2})]y^{2} = 0 \&\Rightarrow y^{2} = 0 \quad or \quad y = 0.\end{{align*}}\]

Step 4 - Get \(x^{2}\) from the first Equation

Substitute \(y^{2} = 0\) into the first equation to get the value of \(x^{2}\):\[x^{2} = \frac{3y^{2} - 7x^{2}y^{2}}{1} = 3(0)(1 - 7x(0)) = 0.\]

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to simplify or solve them. This is a critical skill in algebra which helps make equations easier to understand and work with. In the problem at hand, once the fractions are removed from the system of equations, algebraic manipulation helps in isolating the variables. This means we can focus on one part of the equation to make further calculations simpler. By using this method, variables are rearranged to be on one side of the equation, turning complex equations into solvable forms.

• This step helps us set up the equations for the substitution method.
• It forms a crucial part by presenting the equations in a way that aligns with both the elimination or substitution methods.
• Algebraic manipulation prepares the equations for further simplification steps that might involve solving for one variable.

Understanding algebraic manipulation allows students to start putting "pieces" together like a puzzle, eventually revealing the answer once all parts have been rearranged effectively.

Eliminating Fractions

Eliminating fractions can make the process of solving equations much simpler. Fractions tend to complicate calculations, so removing them first can result in a much clearer path to the solution. In the original exercise, each equation's terms are coefficients over squares of variables which act as fractions.

• To eliminate these fractions, find the Lowest Common Multiple (LCM) of the denominators involved. This step prepares equations for easier manipulation.
• In this specific exercise, the LCM of \(x^2\) and \(y^2\) is \(x^2y^2\). Therefore, multiplying through by \(x^2y^2\) removes fractions completely.
• As a result, the equations turn into simple polynomial form, which is much easier to work with.

By removing fractions, equations immediately become less intimidating and more straightforward—a step that sets the stage for further algebraic methods like substitution.

Substitution Method

The substitution method is a strategic approach to solving systems of equations. It involves transforming one equation so it expresses one variable in terms of the other, allowing for direct insertion into the remaining equations. Once we remove fractions and manipulate equations, we apply substitution.

• In our task, once an equation is solved for \(x^2\), it is substituted back in another equation to find the value of \(y^2\).
• This method helps in reducing the number of variables being dealt with in given equations, making it much more manageable.
• After substitution, if the equation reduces to something simple such as \(y^2 = 0\), we know one variable's value immediately.

The substitution method cuts down on solving multi-variable systems independently, enabling a focused and systematic approach to finding each variable in turn. This method is particularly useful when you encounter systems rich with variables needing simplification.